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Abstract

Conflict management is a hot research topic in Dempster–Shafer theory which is used to avoid the counterintuition problem of combination results. In the present conflict management methods, conflicting evidence is assigned to smaller weight to reduce its influence on the combination result. However, these methods will be disabled when similarity collision occurs. In this article, a new conflict management method based on similarity and Basic Probability Assignment-Matrix is proposed; in the proposed method, similarity collision is diminished by computing the index of each element in evidence, and then, a more reasonable evidence weight can be determined in this way. In the end, two experiments are set to compare the results of several different methods, and the results illustrate that the conflicting evidence will be assigned to a smaller weight in our method than in others.

Keywords

Dempster–Shafer theory similarity of evidence conflict management evidence conflict basic probability assignment

Introduction

Dempster–Shafer (D-S) theory¹ is an effective tool to make a decision from several answers with ambiguity. To each answer, the probability that it must be true is denoted as Bel and the probability that it cannot be false is denoted as Pl. Bel is also marked as BPA (basic probability assignment) or mass function m. All the BPAs of a same question will constitute a D-S evidence. A set of evidence can be combined into a new piece of evidence by combination rules, and the largest BPA in the combination result represents the decision. Traditionally, the biggest BPA in BOE (Body of Evidence) is the answer supported by the evidence. Based on the attributes above, evidence theory is widely applied in reliable analysis,^2–5 relationship computing,⁶ making decision,^7–10 and optimal problems.^11–13

However, the evidence we gathered may be not accurate and even the answer supported by each evidence may be different.¹⁴ This phenomenon is the conflict of evidence,^15,16 and the defects of combining conflicting evidence directly mainly lies on:¹⁷

Combination a set of conflicting evidence may assign 100% probability to a BPA which is counterintuitive.

The combination result may be totally wrong due to a highly conflicting evidence.

The BPA equal to zero will always be zero in combination whatever the other BOES.

To overcome the defects above, many scholars proposed their conflict management methods to control the influence of conflict evidence by assigning weight to each evidence. Based on evidence similarity values, weight of each evidence is determined. Wen et al.¹⁸ proposed a similarity calculation method based on computing the cosine value between evidence. Zhao et al.⁶ proposed another conflict management method based on computing the intersection part of evidence. In addition, similarity of evidence can be obtained by computing the distance of evidence.

Cuzzlion¹⁹ explained the meaning of evidence distance in geometric view which stands for the process that transforms the difference between evidence into the distance of points in geometric space. Jousselme et al.²⁰ proposed an evidence distance calculation method based on the difference of BOEs and the number of elements in BPA. Sunberg and Rogers²¹ modified Jousselme’s distance by comparing the difference of the biggest and the smallest BPA in evidence.

Based on similarity of evidence, Deng et al.²² proposed a conflict management method by transferring similarity values into the weight of evidence. Wang et al.²³ modified Murphy’s combination rule by computing evidence distance. Chin and Fu²⁴ proposed another conflict management method based on evidence distance and mass function. Su et al.²⁵ proposed a combination method for determining dependent evidence based on conflict management. Jiang et al.²⁶ proposed a fault diagnosis method based on managing conflicting evidence. Yang and Han²⁷ proposed an uncertainty measure method based on conflict management. Sankararaman and Mahadevan²⁸ proposed a validation measure method based on conflict management.

But similarity calculation methods used by many scholars are not as faultless as they believe. Similarity calculation is the key in conflict management, and the difference of BOEs should be represented by the difference of similarity values. However, the phenomenon that two different pairs of evidence share a same similarity value is easy to be found, and this phenomenon is the collision of similarity which is mainly caused by the BPA sequence in BOE is not computed, and the exchange of BPA sequence may not change the similarity calculation result.

In this article, a matrix named BPA-Matrix (Basic Probability Assignment-Matrix) that stands for BPA sequence in BOE is introduced. With the introduction of BPA-Matrix, collision of similarity is diminished and weight of evidence becomes more reasonable. The frame of this article is as follows: section “Introduction” introduces D-S theory and the necessity of conflict management. Section “Preliminaries” depicts some important definitions and equations used in conflict management method proposed in this article, and a brief example is given to illustrate the defects in existing conflict management. Section “The method proposed” illustrates the process of the proposed conflict management, and an example is given to illustrate the computing process of BPA-Matrix. Section “Experiments” is the experiment part, and the weight of each evidence determined by several conflict management methods is compared under two different sets of evidence.

Preliminaries

D-S evidence theory

Let Ω be a set containing several exclusive and exhaustive elements: ${H_{1}, H_{2}, H_{3}, \dots, H_{N}}$ , Ω is denoted as the frame of discernment and $P (Ω)$ is a power set where $P (Ω) = 2^{Ω}$ . For any subset of $P (Ω)$ which is marked as A, Bel(A) is the probability that A is true. Bel(A) is also marked as m(A) or BPA, and many BPAs can constitute a BOE

m : (m (A), m (B), m (C), \dots, m (AB), \dots, m (Ω), m (\emptyset))

In the equation above, A, B, and C are single-element focal elements, and AB is multi-element focal element. m(A) or m(B) is BPA of A or B, and m(Ω) is the probability that all the focal elements are uncertain. All the BPAs will constitute a piece of D-S theory evidence which is marked as m. A set of evidence can be combined into a new piece of evidence by combination rule, and the combination rule proposed by Dempster is as follows

m (A) = {\begin{matrix} 0, A = \emptyset \\ \frac{\sum_{A_{i} \cap B_{j} = A} m_{1} (A_{i}) m_{2} (B_{j})}{C}, A \neq \emptyset \end{matrix}

(1)

C = 1 - \sum_{A_{i} \cap B_{j} = \emptyset} m_{1} (A_{i}) m_{2} (B_{j}) or C = \sum_{A_{i} \cap B_{j} \neq \emptyset} m_{1} (A_{i}) m_{2} (B_{j})

(2)

In equations (1) and (2), m is the combination result of m₁ and m₂. Equations (1) and (2) denote the combination rule when two pieces of evidence are combined, and equations (1) and (2) will be transferred into equations (3) and (4) when the number of evidence is larger than two

m (A) = {\begin{matrix} 0, A = \emptyset \\ \frac{\sum_{\cap A_{i} = A} \underset{1 \leq i \leq n}{Π} m_{i} (A_{i})}{C}, A \neq \emptyset \end{matrix}

(3)

C = 1 - \sum_{\cap A_{i} = \emptyset} \underset{1 \leq i \leq n}{Π} m_{i} (A_{i}) or C = \sum \underset{1 \leq i \leq n}{Π} m_{i} (A_{i})

(4)

According to equations (1)–(4), combination rule proposed by Dempster meets the law of commutation and the law of association.⁷ Even though evidence can be combined based on combination rule proposed by Dempster, the combination result may be counterintuitive. A brief example is given in the following section.

Example 1

Assuming m₁ and m₂ are two pieces of evidence under a same frame of discernment, and the BOEs of m₁ and m₂ are given as follows

\begin{matrix} m_{1} : m_{1} (X) = 0.99, m_{1} (Y) = 0.01, m_{1} (Z) = 0.00 \\ m_{2} : m_{2} (X) = 0.00, m_{2} (Y) = 0.01, m_{2} (Z) = 0.99 \end{matrix}

From equation (2), C is obtained as follows: $C = \sum_{A_{i} \cap B_{j} \neq \emptyset} m_{1} (A_{i}) m_{2} (B_{j}) = 0.0001$ .

And, focal elements in combination result can be obtained by equation (1)

\begin{matrix} m (X) = \frac{\sum_{A_{i} \cap B_{j} = X} m_{1} (A_{i}) m_{2} (B_{j})}{C} = \frac{m_{1} (X) m_{2} (X)}{C} = 0.00 \\ m (Y) = \frac{\sum_{A_{i} \cap B_{j} = Y} m_{1} (A_{i}) m_{2} (B_{j})}{C} = \frac{m_{1} (Y) m_{2} (Y)}{C} = 1.00 \\ m (Z) = \frac{\sum_{A_{i} \cap B_{j} = Z} m_{1} (A_{i}) m_{2} (B_{j})}{C} = \frac{m_{1} (Z) m_{2} (Z)}{C} = 0.00 \end{matrix}

The combination result of m₁ and m₂ is $m : m (X) = 0.00, m (Y) = 1.00, m (Z) = 0.00$ .The focal element supported in evidence m₁ is X, and the focal element supported in evidence m₂ is Z, but the focal element supported in the combination result is Y, which is counterintuitive. To overcome the shortage above, conflict management of evidence is proposed by many scholars to avoid a counterintuition combination result by assigning weight to each evidence, and the weight of conflicting evidence is smaller than the weight of un-conflicting evidence. The weight is obtained by similarity calculation, and the more similar the evidence is toward the others, the less conflict it causes.

Similarity of evidence

Similarity calculation

As described in section “Introduction,” the difference between evidence can be transferred into a value by similarity calculation. And, similarity calculation method proposed by Wen et al.¹⁸ is defined in Definition 1.

Definition 1 (evidence similarity proposed by Wen et al.)

Assuming that m₁ and m₂ are two pieces of evidence under a same frame of discernment, the similarity value between m₁ and m₂ is as follows

si m_{wen} (m_{1}, m_{2}) = \frac{m_{1} \cdot m_{2}^{T}}{| | m_{1} | | \cdot | | m_{2} | |}

(5)

$| | m_{1} | |$ and $| | m_{2} | |$ are the norms of m₁ and m₂, and $m_{1} \cdot m_{2}^{T}$ is the Cartesian product of m₁ and m₂. Similarity calculation method proposed by Wen et al. represents a series of methods which compute similarity directly. In addition, some scholars proposed their similarity calculation methods based on evidence distance. Jousselme et al.²⁰ proposed four limits which most evidence distance calculation method should follow:

Nonnegativity: $d (m_{1}, m_{2}) \geq 0$ ;

Nondegeneracy: $d (m_{1}, m_{2}) = 0 \Leftrightarrow m_{1} = m_{2}$ ;

Symmetry: $d (m_{1}, m_{2}) = d (m_{2}, m_{1})$ ;

Triangle: $d (m_{1}, m_{2}) \leq d (m_{1}, m_{3}) + d (m_{3}, m_{2})$ .

$d (m_{1}, m_{2})$ is the distance of evidence m₁ and m₂. Based on the limits above, many scholars proposed their evidence distance calculation methods and method proposed by Jousselme is defined in the following section.

Definition 2 (evidence distance proposed by Jousselme et al.)

Assuming that m₁ and m₂ are two pieces of evidence under a same frame of discernment, the distance between m₁ and m₂ is as follows

d_{i, j} = \sqrt{\frac{1}{2} {({\vec{m}}_{i} - {\vec{m}}_{j})}^{T} D ({\vec{m}}_{i} - {\vec{m}}_{j})}

(6)

${\vec{m}}_{i}$ and ${\vec{m}}_{j}$ are the vector form of m_i and m_j, D is a matrix defined as $D (A, B) = | A \cap B | / | A \cup B |$ where A and B are focal elements. Based on the distance of evidence, similarity of evidence can be obtained as follows

si m_{Jou} (m_{i}, m_{j}) = 1 - d_{i, j}

(7)

Collision of similarity

Similarity calculation is a key process in conflict management, but collision of similarity may occur in both the methods proposed by Jousselme et al. and Wen et al. A brief example is given in the following section.

Example 2 (similarity collision)

Assuming that m₁, m₂, and m₃ are three pieces of evidence under a same frame of discernment, the BOE of each evidence is shown as follows

\begin{array}{l} m_{1} : m_{1} (A) = 0.3, m_{1} (B) = 0.2, m_{1} (C) = 0.1, m_{1} (A C) = 0.4 \\ m_{2} : m_{2} (A) = 0.1, m_{2} (B) = 0.2, m_{2} (C) = 0.3, m_{2} (A C) = 0.4 \\ m_{3} : m_{3} (A) = 0.2, m (B)_{3} = 0.2, m (C)_{3} = 0.2, m_{3} (A C) = 0.4 \end{array}

Based on equation (5), similarity value between m₁ and m₃ in Wen et al.’s method is as follows

\begin{array}{l} s i m_{w e n} (m_{1}, m_{3}) = \frac{m_{1} \cdot m_{3}^{T}}{| | m_{1} | | \cdot | | m_{3} | |} \\ = \frac{(0.3, 0.2, 0.1, 0.4) \cdot {(0.2, 0.2, 0.2, 0.4)}^{T}}{| | (0.3, 0.2, 0.1, 0.4) | | \cdot | | 0.2, 0.2, 0.2, 0.4 | |} = \frac{0.28}{0.29} = 0.97 \end{array}

And, similarity value between m₂ and m₃ in Wen et al.’s method is as follows

\begin{array}{l} s i m_{w e n} (m_{2}, m_{3}) = \frac{m_{2} \cdot m_{3}^{T}}{| | m_{2} | | \cdot | | m_{3} | |} \\ = \frac{(0.1, 0.2, 0.3, 0.4) \cdot {(0.2, 0.2, 0.2, 0.4)}^{T}}{| | (0.1, 0.2, 0.3, 0.4) | | \cdot | | 0.2, 0.2, 0.2, 0.4 | |} = \frac{0.28}{0.29} = 0.97 \end{array}

According to equation (6), the distance of m₁ and m₃ is as follows

d_{1, 3} = \sqrt{\frac{1}{2} {({\vec{m}}_{1} - {\vec{m}}_{3})}^{T} D ({\vec{m}}_{1} - {\vec{m}}_{3})} = \sqrt{\frac{1}{2} {(0.1, 0, - 0.1, 0)}^{T} [\begin{matrix} 1 & 0 & 0 & \frac{1}{2} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2} \\ \frac{1}{2} & 0 & \frac{1}{2} & 1 \end{matrix}] (0.1, 0, - 0.1, 0)} = 0.1

From equation (7), similarity value between m₁ and m₃ based on distance is as follows

si m_{Jou} (m_{1}, m_{3}) = 1 - d_{1, 3} = 1 - 0.1 = 0.9

And, distance of m₂ and m₃ is as follows

d_{2, 3} = \sqrt{\frac{1}{2} {({\vec{m}}_{2} - {\vec{m}}_{3})}^{T} D ({\vec{m}}_{2} - {\vec{m}}_{3})} = \sqrt{\frac{1}{2} {(- 0.1, 0, 0.1, 0)}^{T} [\begin{matrix} 1 & 0 & 0 & \frac{1}{2} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{1}{2} \\ \frac{1}{2} & 0 & \frac{1}{2} & 1 \end{matrix}] (- 0.1, 0, 0.1, 0)} = 0.1

And, similarity between m₂ and m₃ based on distance is as follows

si m_{Jou} (m_{2}, m_{3}) = 1 - d_{2, 3} = 1 - 0.1 = 0.9

According to similarity calculation results above, sim_wen(m₁,m₃) = sim_wen(m₂,m₃) and sim_Jou(m₁,m₃) =sim_Jou(m₂,m₃) but m₁ ≠ m₂. The difference between m₂ and m₃ lies on the exchange of A and C. Since BPA sequence in BOE is not computed in similarity calculation, the alternation of BPA sequence may not change the similarity calculation result. Similarity of evidence is the base of conflict management, and the collision of similarity may lead to the deviation of the weight assigned to evidence. Before introducing the conflict management method in this article, further steps in previous conflict management only based on similarity are shown in section “Conflict management only based on similarity.”

Conflict management only based on similarity

Definition 3 (support of evidence)

Assuming $m_{1}, m_{2}, m_{3}, \dots, m_{n}$ is a set of evidence under a same frame of discernment, and $sim (m_{i}, m_{1}), sim (m_{i}, m_{2}), sim (m_{i}, m_{3}), \dots, sim (m_{i}, m_{n})$ are similarity values among $m_{1}, m_{2}, m_{3}, \dots, m_{n}$ , the support value $Su p_{i}$ of $m_{i}$ is as follows

Su p_{i} = \sum_{j = 1, j \neq i}^{n} sim (m_{i}, m_{j})

(8)

Support of evidence can be obtained by equation (8), but the range of evidence support is not [0, 1]. Note that support of evidence is different with the focal element supported by a piece of evidence. To realize the weight determination of each evidence, a new parameter named evidence credit is defined in the following section.

Definition 4 (credit of evidence)

Assuming that $m_{1}, m_{2}, m_{3}, \dots, m_{n}$ is a set of evidence under a same frame of discernment, and $Su p_{1}, Su p_{2}, Su p_{3}, \dots, Su p_{n}$ are evidence support values of $m_{1}, m_{2}, m_{3}, \dots, m_{n}$ , the credit value $Cre d_{i}$ of $m_{i}$ is as follows

Cre d_{i} = \frac{Su p_{i}}{\sum_{j = 1}^{n} Su p_{j}}

(9)

Evidence credit is the weight of evidence in traditional conflict management. Referring to Example 2, similarity of evidence may collide, and conflict management only based on similarity will be invalid. To overcome the shortage, conflict management based on similarity and a matrix named BPA-Matrix which depicts the BPA sequence in BOE is proposed.

The method proposed

According to Example 2, conflict management only based on similarity of evidence is not sufficient. And, BPA sequence needs to be computed in conflict management to diminish the collision of similarity. The frame of conflict management proposed in this article is shown in Figure 1.

Figure 1.

Frame of the proposed conflict management method.

There are three mainly parts in our method: BPA-Factor calculation, evidence credit calculation, and the combination of them. In the beginning of BPA-Factor calculation part, BPA sequence in BOE is converted into a matrix named BPA-Matrix, and the difference of each BPA-Matrix is transferred into a BPA-Factor which is marked as F in the end of this part. Evidence credit calculation part acts as the traditional conflict management, and the evidence credit based on similarity is computed. In the last part, union support is defined as the fusion of BPA-Factor and evidence credit, and union credit realizes the determination of each evidence weight based on union support.

BPA-Factor calculation

Assuming that D₁ and D₂ are two matrixes, the equation D₃ = D₁ × D₂ can be considered as the amplifier and exchange on columns of D₁ or lines of D₂. The evidence can be considered as a single line matrix D₁, and the BOE with BPA sorted by the size of each BPA can be considered as another single line matrix D₃. With D₁ and D₃ being settled, D₂ can be obtained by Definition 5.

Definition 5 (BPA-Matrix of evidence)

BPA-Matrix is a matrix that BPA sequence can be sorted by multiplying it. BPA-Matrix is marked as BMatrix in this article. The dimension of BMatrix is $2^{Ω} \times 2^{Ω}$ where Ω is the frame of discernment. The BMatrix of evidence m can be obtained by the following steps:

Set a new array R that contains all subsets of P(Ω), the array belongs to all evidence and it is generated only once;

Expand the BOE of m by setting BPA to 0 which exists in R, but not in m, and the expanded BOE is marked as v;

To each BPA in v, mark its index in R as i_BPA, just as R[i_BPA] = BPA;

A new array t is formed by sorting all BPAs in v from big to small and mark the index of each BPA in t as j_BPA;

Create a matrix BMatrix with the dimension 2^Ω × 2^Ω. For each BPA in v, set $BMatri x_{i_{BPA}, j_{BPA}} = 1$ and the other elements in BMatrix as 0.

A brief example is given in the following section to illustrate the calculation process of BPA-Matrix.

Example 3 (BPA-Matrix calculation)

Assuming that m is a piece of evidence under the frame of discern Ω: {A,B,C}, and the BOE of m is as follows

m : m (A) = 0.3, m (B) = 0.2, m (AC) = 0.5

First, we need to specify an array R that contains all subsets of Ω

R = (A, B, C, AB, AC, BC, ABC, \emptyset)

Second, v is obtained by expanding m

\begin{matrix} m : m (A) = 0.3, m (B) = 0.2, m (AC) = 0.5 \Rightarrow \\ ν : ν (A) = 0.3, ν (B) = 0.2, ν (C) = 0, ν (AB) = 0, ν (AC) = 0.5, ν (BC) = 0, ν (ABC) = 0, ν = (\emptyset) = 0 \end{matrix}

To each BPA in v, mark its index in R as i_BPA

i_{A} = 1, i_{B} = 2, i_{C} = 3, i_{AB} = 4, i_{AC} = 5, i_{BC} = 6, i_{ABC} = 7, i_{\emptyset} = 8

A new array t can be obtained by sorting each BPA in v from big to small

t = (t (AC) = 0.5, t (A) = 0.3, t (B) = 0.2, t (C) = 0, t (AB) = 0, t (BC) = 0, t (ABC) = 0, t (\emptyset) = 0)

Here, we get a new index j_BPA of each BPA in t

j_{AC} = 1, j_{A} = 2, j_{B} = 3, j_{C} = 4, j_{AB} = 5, j_{BC} = 6, j_{ABC} = 7, j_{\emptyset} = 8

To each BPA in v, set $BMatri x_{i_{BPA}, j_{BPA}} = 1$ , and the other elements in BMatrix = 0

BMatrix = [\begin{matrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}]

BMatrix is the BPA-Matrix of to m. And, we can find BPAs in BOE can be sorted by multiplying the BPA-Matrix of it, just as the equation below

(0.3, 0.2, 0, 0, 0.5, 0, 0, 0) \times [\begin{matrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] = (0.5, 0.3, 0.2, 0, 0, 0, 0, 0)

Each evidence is mapped into a BMatrix by Definition 5, and a new matrix AMatrix can be obtained as follows

AMatrix = \frac{1}{n} \sum_{i = 1}^{n} BMatri x_{i}

(10)

n is the number of evidence, AMatrix is not only an overview of all BMatrix but also a standard in computing the difference of each BMatrix. The difference can be marked as MMatrix which is defined as follows

MMatri x_{i} = BMatri x_{i} - AMatrix

(11)

Each evidence is mapped into a BMatrix and a MMatrix. However, MMatrix is a matrix and it is inconvenient to be combined with other parameters. To overcome the shortage, BPA-Factor marked as F is introduced to realize the process that transforming MMatrix into a numerical value which is defined in the following section.

Definition 6 (BPA-Factor of evidence)

Assume that $m_{1}, m_{2}, m_{3}, \dots, m_{n}$ is a set of evidence under a same frame of discernment. And, $MMatri x_{1}, MMatri x_{2}, MMatri x_{3}, \dots, MMatri x_{n}$ are MMatrixes belonging to $m_{1}, m_{2}, m_{3}, \dots, m_{n}$ . The BPA-Factor $F_{i}$ belonging to $m_{i}$ is as follows

F_{i} = \frac{n \times e^{- ‖ MMatri x_{i} ‖}}{\sum_{j = 1}^{n} e^{- ‖ MMatri x_{j} ‖}}

(12)

$| | MMatri x_{i} | |$ is the norm of $MMatri x_{i}$ , and the larger $| | MMatri x_{i} | |$ is, the bigger the difference of two BPA sequences is. Note that the value range of BPA-Factor is not [0, 1]. Based on the previous steps, difference of each BPA sequence is represented as a numerical value F, and the following operation is similarity calculation which is shown in section “Evidence credit calculation based on similarity.”

Evidence credit calculation based on similarity

According to section “Similarity calculation,” similarity calculation of evidence is realized by two kinds of methods. The first kind is based on computing the difference between two pieces of evidence directly, and the second is based on evidence distance. The similarity calculation used in this article is the method based on evidence distance which is marked as $sim (m_{i}, m_{j})$ , and we can find $sim (m_{i}, m_{j}) = si m_{Jou} (m_{i}, m_{j})$ easily.

Based on equation (8), evidence support can be obtained as follows

Su p_{i} = \sum_{j = 1, j \neq i}^{n} sim (m_{i}, m_{j})

With evidence support is obtained, evidence credit can be obtained based on equation (9)

Cre d_{i} = \frac{Su p_{i}}{\sum_{j = 1}^{n} Su p_{j}}

Combining BPA-Factor and evidence credit

According to Example 2 in section “Preliminaries,” conflict management only based on similarity is not sufficient, and BPA-Matrix is introduced to diminish the collision of similarity. To realize the combination of BPA-Matrix calculation and similarity calculation, a new parameter named union support is introduced. The definition of union support is described in the following section.

Definition 7 (union support of evidence)

Assuming that $m_{1}, m_{2}, m_{3}, \dots, m_{n}$ is a set of evidence under a same frame of discernment, $Cre d_{1}, Cre d_{2}, Cre d_{3}, \dots, Cre d_{n}$ and $F_{1}, F_{2}, F_{3}, \dots, F_{n}$ are evidence credits and BPA-Factors of $m_{1}, m_{2}, m_{3}, \dots, m_{n}$ , the union support $USu p_{i}$ belonging to $m_{i}$ is as follows

USu p_{i} = F_{i} \times Cre d_{i}

(13)

Note that the value range of union support is not [0, 1]. To realize the weight determination of each evidence, another process is needed to transform union support into a value with value range [0, 1]. The value is marked as union credit and is defined in the following section.

Definition 8 (union credit of evidence)

Assuming that $m_{1}, m_{2}, m_{3}, \dots, m_{n}$ is a set of evidence under the same frame of discernment, and $USu p_{1}, USu p_{2}, USu p_{3}, \dots, USu p_{n}$ are union support values belonging to $m_{1}, m_{2}, m_{3}, \dots, m_{n}$ , and the union credit $UCre d_{i}$ belonging to $m_{i}$ is as follows

UCre d_{i} = \frac{USu p_{i}}{\sum_{j = 1}^{n} USu p_{j}}

(14)

Union credit is the weight of evidence in conflict management. Compared with previous conflict management only based on similarity, conflict management in this article is sensitive to BPA sequence in BOE. The exchange of BPA sequence will lead to the alternation of union credit and similarity collision is diminished. In addition, equation (14) can be expressed as equation (15) using primary similarity and BPA-Matrix

UCre d_{i} = \frac{\frac{n \times e^{- ‖ BMatri x_{i} - \frac{1}{n} \sum_{t = 1}^{n} BMatri x_{t} ‖}}{\sum_{k = 1}^{n} e^{- ‖ BMatri x_{k} - \frac{1}{n} \sum_{t = 1}^{n} BMatri x_{t} ‖}} \times \frac{\sum_{x = 1, x \neq i}^{n} sim (m_{i}, m_{x})}{\sum_{j = 1}^{n} \sum_{y = 1, y \neq j}^{n} sim (m_{j}, m_{y})}}{\sum_{v = 1}^{n} (\frac{n \times e^{- ‖ BMatri x_{v} - \frac{1}{n} \sum_{t = 1}^{n} BMatri x_{t} ‖}}{\sum_{u = 1}^{n} e^{- ‖ BMatri x_{u} - \frac{1}{n} \sum_{t = 1}^{n} BMatri x_{t} ‖}} \times \frac{\sum_{x = 1, x \neq v}^{n} sim (m_{v}, m_{x})}{\sum_{j = 1}^{n} \sum_{y = 1, y \neq j}^{n} sim (m_{j}, m_{y})})}

(15)

Experiments

In this section, two experiments are given to compare the effect of the four representative methods for conflict management.

Experiment 1

Suggest that $m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$ are four D-S evidence under a same frame of discernment $Ω : {A, B, C}$ and the BOE of each evidence is given as follows

\begin{matrix} m_{1} : m_{1} (A) = 0.7, m_{1} (B) = 0.2, m_{1} (C) = 0.1 \\ m_{2} : m_{2} (A) = 0.1, m_{2} (B) = 0.2, m_{2} (C) = 0.7 \\ m_{3} : m_{3} (A) = 0.4, m_{3} (B) = 0.2, m_{3} (C) = 0.4 \\ m_{4} : m_{4} (A) = 0.8, m_{4} (B) = 0.15, m_{4} (C) = 0.05 \end{matrix}

The first attribution to be compared is the similarity value in different methods which is shown below. From the similarity calculation results in Tables 1 –3, it is obvious that the similarity values that m₁ and m₂ towards m₃ are same; however, m₁ is different from m₂. This phenomenon is the collision of similarity. In our method, the collision can be detected by BPA-Matrix and represented as BPA-Factor in this article. The BPA-Factor of each evidence is shown in Figure 2.

Table 1.

Similarity values in proposed method.

Similarity	$m_{1}$	$m_{2}$	$m_{3}$	$m_{4}$
$m_{1}$	1	0.4	0.7	0.913397
$m_{2}$	0.4	1	0.7	0.323612
$m_{3}$	0.7	0.7	1	0.622508
$m_{4}$	0.913397	0.323612	0.622508	1

Table 2.

Similarity values in Zhao et al.’s method.

Similarity	$m_{1}$	$m_{2}$	$m_{3}$	$m_{4}$
$m_{1}$	1	0.244948	0.542161	0.768622
$m_{2}$	0.244948	1	0.542161	0.187311
$m_{3}$	0.542161	0.542161	1	0.528957
$m_{4}$	0.768622	0.187311	0.528957	1

Table 3.

Similarity values in Wen et al.’s method.

Similarity	$m_{1}$	$m_{2}$	$m_{3}$	$m_{4}$
$m_{1}$	1	0.333333	0.816496	0.992908
$m_{2}$	0.333333	1	0.816496	0.241969
$m_{3}$	0.816496	0.816496	1	0.756205
$m_{4}$	0.992908	0.241969	0.756205	1

Figure 2.

BPA-Factor of each evidence.

Based on similarity values and BPA-Factor above, evidence support or union support can be obtained and is shown in Table 4.

Table 4.

Comparison of evidence supports.

Method	Evidence
Method	m ₁	m ₂	m ₃	m ₄
Wen et al.	2.142738	1.391799	2.389198	1.991083
Wang et al.	2.013397	1.423613	2.022508	1.859518
Zhao et al.	1.555733	0.974422	1.613279	1.484891
Method proposed	0.361647	0.133307	0.189387	0.334007

Support calculation of evidence is not the last step in conflict management, and evidence credit or union credit in this article can be obtained by evidence support. Evidence credit or union credit is shown in Table 5.

Table 5.

Comparison of evidence credits.

Method	Evidence
Method	m ₁	m ₂	m ₃	m ₄
Wen et al.	0.896844	0.582538	1.00000	0.833369
Wang et al.	0.275091	0.194508	0.276335	0.254066
Zhao et al.	0.276411	0.173128	0.286639	0.263825
Method proposed	0.355131	0.130905	0.185975	0.327989

According to the tables above, the proportion of each evidence credit value in the sum of all the credit values is given in Figure 3, which is proportional to the weight of evidence in the final combination.

Figure 3.

Comparison of proportion of evidence credit.

As m₂ or m₃ is conflicting evidence, the proportions of its credit values should be as small as possible. Oppositely, m₁ or m₄ should have larger credit proportion for it is not conflicting evidence. From the experiment result, it is obvious that the proportion of evidence credit value computed by our method is the most reasonable.

Experiment 2

Example 2 depicts a more general case that evidence is gathered from the real world, and the collision of similarity may occur or not. We gathered a set of evidence from five temperate sensors as m₁, m₂, m₃, m₄, and m₅, and the BOE of each evidence is shown as follows

\begin{matrix} m_{1} : m_{1} (A) = 0.4, m_{1} (B) = 0.15, m_{1} (C) = 0.45 \\ m_{2} : m_{2} (B) = 0.9, m_{2} (C) = 0.1 \\ m_{3} : m_{3} (A) = 0.68, m_{3} (B) = 0.07, m_{3} (AC) = 0.25 \\ m_{4} : m_{4} (A) = 0.45, m_{4} (B) = 0.1, m_{4} (AC) = 0.45 \\ m_{5} : m_{5} (A) = 0.59, m_{5} (B) = 0.1, m_{5} (C) = 0.01, \\ m_{5} (AC) = 0.3 \end{matrix}

The similarity values of evidence are shown in Tables 6 –8.

Table 6.

Similarity values in proposed method.

Similarity	$m_{1}$	$m_{2}$	$m_{3}$	$m_{4}$	$m_{5}$
$m_{1}$	1	0.35	0.608018	0.660884	0.648432
$m_{2}$	0. 35	1	0.172715	0.221379	0.213806
$m_{3}$	0.608018	0.172715	1	0.845404	0.938355
$m_{4}$	0.660884	0.221379	0.845404	1	0.900752
$m_{5}$	0.648432	0.213806	0.938355	0.900752	1

Table 7.

Similarity values in Zhao et al.’s method.

Similarity	$m_{1}$	$m_{2}$	$m_{3}$	$m_{4}$	$m_{5}$
$m_{1}$	1	0.240134	0.420363	0.308431	0.396422
$m_{2}$	0.240134	1	0.077599	0.117835	0.116874
$m_{3}$	0.420363	0.077599	1	0.621383	0.692199
$m_{4}$	0.308431	0.117835	0.621383	1	0.625076
$m_{5}$	0.396422	0.116874	0.692199	0.625076	1

Table 8.

Similarity values in Wen et al.’s method.

Similarity	$m_{1}$	$m_{2}$	$m_{3}$	$m_{4}$	$m_{5}$
$m_{1}$	1	0.320357	0.625506	0.487843	0.615070
$m_{2}$	0.320357	1	0.095582	0.154281	0.150106
$m_{3}$	0.625506	0.095582	1	0.907443	0.991596
$m_{4}$	0.487843	0.154281	0.907443	1	0.951816
$m_{5}$	0.615070	0.150106	0.991596	0.951816	1

According to three tables below collision of similarity does not occur. However, BPA sequences in BOEs are different, and the biggest focal element in each evidence may be different. This kind of difference can be represented by BPA-Factor F, and Figure 4 is the comparison of each BPA-Factor.

Figure 4.

BPA-Factor of each evidence.

Referring to the giving evidence, we can find that m₁ and m₂ are conflicting evidence. BPA-Factor of them is smaller than that of m₃, m₄ and m₅. Based on BPA-Factor and similarity, evidence support of each evidence in each method can be obtained and is shown in Table 9.

Table 9.

Comparison of evidence supports.

Method	Evidence
Method	m ₁	m ₂	m ₃	m ₄	m ₅
Wen et al.	2.048777	0.720326	2.620129	2.501383	2.708589
Wang et al.	2.267333	0.957901	2.564492	2.628419	2.701347
Zhao et al.	1.365350	0.552443	1.811545	1.672726	1.830572
Method proposed	0.130029	0.049049	0.291588	0.298856	0.307148

Based on evidence support, evidence credit can be obtained by Definition 4. And, the comparison of each evidence credit is shown in Table 10.

Table 10.

Comparison of evidence credits.

Method	Evidence
Method	m ₁	m ₂	m ₃	m ₄	m ₅
Wen et al.	0.756400	0.265941	0.967341	0.923500	1.000000
Wang et al.	0.203906	0.086146	0.230630	0.236379	0.242937
Zhao et al.	0.188776	0.076382	0.250468	0.231274	0.253098
Method proposed	0.120769	0.045556	0.270823	0.277574	0.285275

Based on the values in tables above, the proportion of each evidence credit is shown in Figure 5.

Figure 5.

Comparison of proportion of evidence credit.

Within the giving evidence, m₁ and m₂ are conflicting evidence. The proportions of their credits are the smallest in our method. And, the proportions of m₃, m₄, m₅ computed by our method are the largest. It is obvious that the proportion of evidence credit value computed by our method is the most reasonable.

Two experiments above describe two kinds of cases that similarity collision occurs or not. As BPA-Matrix is an extra step, there should be more time cost in our proposed scheme when compared with the methods only based on similarity. When the number of evidence is n, we need to compute similarity value for n² times, but BPA-Matrix is computed only n times, that is, compared with the time cost of similarity calculation, the time cost of BPA-Matrix computing is tiny. As similarity calculation method used in conflict management method proposed by Wang et al. and this article is same, a comparison of time cost in processing same evidence scale on a same platform (CPU: i7 4710, GPU: GTX960, RAM: 16G, ROM: 250G SSD) is shown in Figure 6.

Figure 6.

Time cost in two management methods.

From Figure 6, time costs by two methods are almost same when handling with same evidence. However, the weight determination by our conflict management method is more reasonable. Based on the comparisons above, it shows that our scheme is a better conflict management method for assigning more reasonable weight to evidence with very tiny performance loss.

Conclusion

In this article, a new conflict management method based on similarity and BPA-Matrix is proposed. Compared with the previous methods, similarity collision is diminished, and the weight determination is more reasonable. In more general cases, the weight determination by our method is more reasonable no matter whether similarity collision occurs or not. In the end, we compared the time cost of our scheme and the method of Wang et al. which is based on similarity. The experiment result shows that the time cost in the two methods is almost same; however, the proportion of evidence credit value in our scheme is more reasonable, which means that the determination of evidence weight is more reasonable. The determination of weights assigned to BPA-Matrix and similarity in conflicting management will be a new point in our future research.

Footnotes

Academic Editor: Nageswara SV Rao

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work was sponsored by National Natural Science Foundation of China grant no. 61370220, Plan For Scientific Innovation Talent of Henan Province grant no. 174200510011, Program for Innovative Research Team (in Science and Technology) in University of Henan Province grant no. 15IRTSTHN010, Program for Henan Province Science and Technology grant no. 142102210425, Natural Science Foundation of Henan Province grant no. 162300410094, Project of the Cultivation Fund of Science and Technology Achievements of Henan University of Science and Technology grant no. 2015BZCG01.

References

Dempster

Upper and lower probabilities induced by a multi-valued mapping. Ann Math Stat 1967; 38: 325–339.

Liu

You

Fan

. Failure mode and effects analysis using D numbers and grey relational projection method. Expert Syst Appl 2014; 41(10): 4670–4679.

Jiang

Xie

Wei

. A modified method for risk evaluation in failure modes and effects analysis of aircraft turbine rotor blades. Adv Mech Eng 2016; 8(4): 1–16.

Mahadevan

. Dependence assessment in human reliability analysis using evidence theory and AHP. Risk Anal 2015; 35: 1296–1316.

Yuan

Xiao

Fei

. Modeling sensor reliability in fault diagnosis based on evidence theory. Sensors 2016; 16(1): 113.

Zhao

Zuo

Tian

. A method for assessment of trust relationship strength based on the improved D-S evidence theory. Chin J Comput 2014; 37(4): 873–884.

Beynon

Curry

Morgan

The Dempster–Shafer theory of evidence: an alternative approach to multicriteria decision modelling. Omega 2000; 28(1): 37–50.

Leung

N-N

J-H.

An integrated information fusion approach based on the theory of evidence and group decision-making. Inform Fusion 2013; 14(4): 410–422.

Jiang

Yang

Luo

. Determining basic probability assignment based on the improved similarity measures of generalized fuzzy numbers. Int J Comput Commun 2015; 10(3): 333–347.

10.

Jiang

Luo

Qin

. An improved method to rank generalized fuzzy numbers with different left heights and right heights. J Intell Fuzzy Syst 2015; 28: 2343–2355.

11.

Deng

Mahadevan

Zhou

Vulnerability assessment of physical protection systems: a bio-inspired approach. Int J Unconv Comput 2015; 11(3–4): 227–243.

12.

Deng

Liu

Zhou

An improved genetic algorithm with initial population strategy for symmetric TSP. Math Probl Eng 2015; 2015: 212794.

13.

Gao

Liu

. Limited-information particle swarm optimization. Appl Math Comput 2015; 268: 832–838.

14.

Yager

RR.

On the Dempster-Shafer framework and new combination rules. Inform Sciences 1987; 41(2): 93–137.

15.

Zadeh

LAA

. Simple view of the Dempster-Shafer theory of evidence and its implication for the rule of combination. AI Mag 1986; 7: 85–90.

16.

Deng

Generalized evidence theory. Appl Intell 2015; 43(3): 530–543.

17.

Huang

Gao

Information fusion technology and application. Beijing, China: National Defense Industry Press, 2010.

18.

Wen

Wang

. Fuzzy information fusion algorithm of fault diagnosis based on similarity measure of evidence. In: Proceedings of the international symposium on neural networks, Beijing, China, 24–28 September 2008, pp. 506–515. Berlin: Springer.

19.

Cuzzolin

A geometric approach to the theory of evidence. Decis Support Syst 2011; 52(1): 133–141.

20.

Jousselme

Grenier

Éloi Bossé . A new distance between two bodies of evidence. Inform Fusion 2001; 2(2): 91–101.

21.

Sunberg

Rogers

A belief function distance metric for orderable sets. Inform Fusion 2013; 14(4): 361–373.

22.

Deng

Shi

Zhu

. Combining belief functions based on distance of evidence. Decis Support Syst 2004; 38(3): 489–493.

23.

Wang

Xiao

Deng

. Weighted evidence combination based on distance of evidence and entropy function. Int J Distrib Sens N 2016; 12(7): 3218784.

24.

Chin

K-S

Weighted cautious conjunctive rule for belief functions combination. Inform Sciences 2015; 325: 70–86.

25.

Mahadevan

Han

. Combining dependent bodies of evidence. Appl Intell 2016; 44: 634–644.

26.

Jiang

Wei

Xie

. An evidential sensor fusion method in fault diagnosis. Adv Mech Eng 2016; 8(3): 1–7.

27.

Yang

Han

A new distance-based total uncertainty measure in the theory of belief functions. Knowl: Based Syst 2016; 94: 114–123.

28.

Sankararaman

Mahadevan

Model validation under epistemic uncertainty. Reliab Eng Syst Safe 2011; 96(9): 1232–1241.

A new conflict management method in Dempster–Shafer theory

Abstract

Keywords

Introduction

Preliminaries

D-S evidence theory

Example 1

Similarity of evidence

Similarity calculation

Definition 1 (evidence similarity proposed by Wen et al.)

Definition 2 (evidence distance proposed by Jousselme et al.)

Collision of similarity

Example 2 (similarity collision)

Conflict management only based on similarity

Definition 3 (support of evidence)

Definition 4 (credit of evidence)

The method proposed

BPA-Factor calculation

Definition 5 (BPA-Matrix of evidence)

Example 3 (BPA-Matrix calculation)

Definition 6 (BPA-Factor of evidence)

Evidence credit calculation based on similarity

Combining BPA-Factor and evidence credit

Definition 7 (union support of evidence)

Definition 8 (union credit of evidence)

Experiments

Experiment 1

Experiment 2

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References